Lines and Planes in Space

Geometry Through Vectors

Vectors give clean descriptions of the basic objects of geometry — lines and planes — in any dimension. The idea: locate a point, then move along direction vectors.

A line through a point $P_0$ (position vector $\mathbf{r}_0$) with direction vector $\mathbf{d}$ is the set of points $$\mathbf{r} = \mathbf{r}_0 + t\,\mathbf{d}, \qquad t \in \mathbb{R}.$$ The parameter $t$ slides you along the line; the direction $\mathbf{d}$ sets the heading. This parametric form works identically in 2D, 3D, or higher, unlike slope-intercept form which is stuck in the plane.

A plane is captured most elegantly by a normal vector $\mathbf{n}$ — a vector perpendicular to the plane. A point $\mathbf{r}$ lies in the plane through $P_0$ with normal $\mathbf{n}$ exactly when the displacement $\mathbf{r} - \mathbf{r}_0$ is perpendicular to $\mathbf{n}$: $$\mathbf{n} \cdot (\mathbf{r} - \mathbf{r}_0) = 0.$$ Expanded, this is the familiar $ax + by + cz = d$, where $(a, b, c)$ is the normal vector — reading the coefficients off a plane's equation gives its normal directly.

These descriptions turn geometry into vector algebra. The angle between two planes is the angle between their normals (via the dot product); two planes are parallel when their normals are parallel and perpendicular when their normals are orthogonal. The distance from a point to a plane, the intersection of a line and a plane, and whether lines are skew all reduce to dot and cross products — the payoff of the vector machinery built in this unit.

Common pitfall: confusing a plane's normal vector with a direction lying in the plane. The normal is perpendicular to the plane — and it is exactly the coefficient vector $(a, b, c)$ in $ax + by + cz = d$, not a vector pointing along the plane. Two planes are parallel when their normals are parallel; using a within-plane direction instead of the normal inverts the geometry.